Dear MHF members,
my problem reads as follows. I have some ideas about some parts of the problem but I would like to share it completely because I find the problem nice.
Problem. Consider the system , where is a constant matrix with only two distinct eigenvalues and Suppose that the Jordan form of has Jordan blocks corresponding to and blocks corresponding to , where is a Jordan block.
- How many linearly independent eigenvectors does have?
- Suppose that is not an eigenvector but satisfies . How many linearly independent solution vectors are there?
- Suppose that the Jordan blocks occur down the diagonal of the in the order stated above. Which matrix elements of are non-zero and what are their values?
Thanks a lot.
bkarpuz
1. The number of regular eigenvectors are , and the remaining are generalized eigenvectors.
2. I feel like it is , but I really need help with this one.
3. The terms of the blocks of the exponential matrix corresponding to will all vanish. For , all the terms except one will vanish. The non-zero term is located at , and its value is .
Right.
2. I feel like it is , but I really need help with this one.
Right. Denote a Jordan block of order k associated to the eigenvalue . Then,
According to the canonical form structure, we verify
3. The terms of the blocks of the exponential matrix corresponding to will all vanish. For , all the terms except one will vanish. The non-zero term is located at , and its value is .
Right. The non zero term corresponds to the block